Arc Consistency and Domain Splitting in CSPs

CPSC 322 – CSP 3

Textbook Poole and Mackworth: § 4.5 and 4.6

Lecturer: Alan Mackworth

October 3, 2012

Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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Constraint Satisfaction Problems (CSPs): Definition

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Definition: A model of a CSP is a possible world that satisfies all constraints.

Definition: A constraint satisfaction problem (CSP) consists of:

• a set of variables V• a domain dom(V) for each variable V V• a set of constraints C

An example CSP:• V = {V1,V2}

– dom(V1) = {1,2,3}– dom(V2) = {1,2}

• C = {C1,C2,C3}– C1: V2 2– C2: V1 + V2 < 5– C3: V1 > V2

Possible worlds for this CSP: {V1=1, V2=1}

{V1=1, V2=2} {V1=2, V2=1} (one model)

{V1=2, V2=2}

{V1=3, V2=1} (another model) {V1=3, V2=2}

Definition: A possible world of a CSP is an assignment of

values to all of its variables.

• Generate and Test:- Generate possible worlds one at a time.- Test constraints for each one.

Example: 3 variables A,B,C

• Simple, but slow: - k variables, each domain size d, c constraints: O(cdk)

Generate and Test (G&T) Algorithms

For a in dom(A)For b in dom(B)

For c in dom(C) if {A=a, B=b, C=c} satisfies all constraints return {A=a, B=b, C=c}

fail

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Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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• Explore search space via DFS but evaluate each constraint as soon as all its variables are bound.

• Any partial assignment that doesn’t satisfy the constraint can be pruned.

• Example: - 3 variables A, B,C, each with domain {1,2,3,4}- {A = 1, B = 1} is inconsistent with constraint A B

regardless of the value of the other variables Fail. Prune!

Backtracking algorithms

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V1 = v1

V2 = v1

V1 = v1

V2 = v2

V1 = v1

V2 = vk

CSP as Graph Searching

V1 = v1

V2 = v1

V3 = v2

V1 = v1

V2 = v1

V3 = v1

{}

V1 = v1 V1 = vk

Check unary constraints on V1If not satisfied PRUNE

Check constraints on V1and V2 If not satisfied PRUNE

Standard Search vs. Specific R&R systems• Constraint Satisfaction (Problems):

– State: assignments of values to a subset of the variables– Successor function: assign values to a ‘free’ variable– Goal test: all variables assigned a value and all constraints

satisfied?– Solution: possible world that satisfies the constraints– Heuristic function: none (all solutions at the same distance

from start)• Planning :

– State– Successor function– Goal test– Solution– Heuristic function

• Inference– State– Successor function– Goal test– Solution– Heuristic function 8

V1 = v1

V2 = v1

V1 = v1

V2 = v2

V1 = v1

V2 = vk

CSP as Graph Searching

V1 = v1

V2 = v1

V3 = v2

V1 = v1

V2 = v1

V3 = v1

{}

V1 = v1 V1 = vk

Check unary constraints on V1If not satisfied PRUNE

Check constraints on V1and V2 If not satisfied PRUNE

Problem?Performance heavily depends on the order in which variables are considered.E.g. only 2 constraints:Vn=Vn-1 and Vn Vn-1

CSP as a Search Problem: another formulation

• States: partial assignment of values to variables• Start state: empty assignment• Successor function: states with the next variable

assigned– Assign any previously unassigned variable– A state assigns values to some subset of variables:

• E.g. {V7 = v1, V2 = v1, V15 = v1}• Neighbors of node {V7 = v1, V2 = v1, V15 = v1}:

nodes {V7 = v1, V2 = v1, V15 = v1, Vx = y} for some variable Vx V \ {V7, V2, V15} and all values ydom(Vx)

• Goal state: complete assignments of values to variables that satisfy all constraints– That is, models

• Solution: assignment (the path doesn’t matter) 10

CSP as Graph Searching• 3 Variables: A,B,C. All with domains = {1,2,3,4}• Constraints: A<B, B<C

• Backtracking relies on one or more heuristics to select which variables to consider next.- E.g. variable involved in the largest number of

constraints: “If you are going to fail on this branch, fail early!”

- Can also be smart about which values to consider first

• This is a different use of the word ‘heuristic’!- Still true in this context

• Can be computed cheaply during the search• Provides guidance to the search algorithm

- But not true anymore in this context• ‘Estimate of the distance to the goal’

• Both meanings are used frequently in the AI literature.• ‘heuristic’ means ‘serves to discover’: goal-oriented.• Does not mean ‘unreliable’!

Selecting variables in a smart way

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Learning Goals for solving CSPs so far

• Verify whether a possible world satisfies a set of constraints i.e. whether it is a model - a solution.

• Implement the Generate-and-Test Algorithm. Explain its disadvantages.

• Solve a CSP by search (specify neighbors, states, start state, goal state). Compare strategies for CSP search. Implement pruning for DFS search in a CSP.

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Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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Can we do better than Search?Key idea • prune the domains as much as possible before

searching for a solution.

• Example: dom(V2) = {1, 2, 3, 4}. V2 2• Variable V2 is not domain consistent. - It is domain consistent once we remove 2 from its

domain.• Trivial for unary constraints. Trickier for k-ary

ones.

Def.: A variable is domain consistent if no value of its domain is ruled impossible by any unary constraints.

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Graph Searching Repeats Work• 3 Variables: A,B,C. All with domains = {1,2,3,4}• Constraints: A<B, B<C• A ≠ 4 is rediscovered 3 times. So is C ≠ 1

- Solution: remove values from A’s domain and C’s, once and for all

• Example: - Two variables X and Y- One constraint: X<Y

X YX< Y

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Def. A constraint network is defined by a graph, with- one node for every variable (drawn as circle)- one node for every constraint (drawn as rectangle)- undirected edges running between variable nodes

and constraint nodes whenever a given variable is involved in a given constraint.

Constraint network: definition

Constraint network: definition

• Whiteboard example: 3 Variables A,B,C– 3 Constraints: A<B, B<C, A+3=C– 6 edges/arcs in the constraint network:

• ⟨A,A<B , B,A<B ⟩ ⟨ ⟩• ⟨B,B<C , C,B<C⟩ ⟨ ⟩• ⟨A, A+3=C , C,A+3=C⟩ ⟨ ⟩

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Def. A constraint network is defined by a graph, with- one node for every variable (drawn as circle)- one node for every constraint (drawn as rectangle)- Edges/arcs running between variable nodes and

constraint nodes whenever a given variable is involved in a given constraint.

A more complicated example• How many variables are there in this constraint

network?

– Variables are drawn as circles

• How many constraints are there?

– Constraints are drawn as rectangles19

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5

9

6

14

5

9

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Arc ConsistencyDefinition:

An arc <x, r(x,y)> is arc consistent if for each value x in dom(X) there is some value y in dom(Y) such that r(x,y) is satisfied.A network is arc consistent if all its arcs are arc consistent.

T FT F

2,5,7 2,3,13A B

A< B/2

Is this arc consistent?

1,2,3 2,3A B

A< B

Not arc consistent: No value in domain of B that satisfies A<B if A=3

Arc consistent: Both B=2 and B=3 have ok values for A (e.g. A=1)

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How can we enforce Arc Consistency?

• If an arc <X, r(X,Y)> is not arc consistent- Delete all values x in dom(X) for which there is no corresponding

value in dom(Y)- This deletion makes the arc <X, r(X,Y)> arc consistent.- This removal can never rule out any models/solutions

• Why?

Run this example: http://cs.ubc.ca/~mack/CS322/AIspace/simple-network.xml in ( (Save to a local file and open file.)

2,3,4 1,2,3X Y

X< Y

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Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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Arc Consistency Algorithm: high level strategy

• Consider the arcs in turn, making each arc consistent

• Reconsider arcs that could be made inconsistent again by this pruning of the domains

• Eventually reach a ‘fixed point’: all arcs consistent

• Run ‘simple problem 1’ in AIspace for an example:

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Which arcs need to be reconsidered?

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every arc Z,c' where c’ c involves Z and X:Z1 c1

Z2 c2

Z3 c3

Yc

THESE

X

Ac4

• When we reduce the domain of a variable X to make an arc X,c arc consistent, which arcs do we need to reconsider?

• You do not need to reconsider other arcs- If arc Y,c was arc consistent before, it will still be arc

consistent- If an arc X,c' was arc consistent before, it will still be arc

consistent- Nothing changes for arcs of constraints not involving X

• Consider the arcs in turn, making each arc consistent

• Reconsider arcs that could be made inconsistent again by this pruning

• DO trace on ‘simple problem 1’ and on ‘scheduling problem 1’, trying to predict

- which arcs are not consistent and - which arcs need to be reconsidered after each removal in

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Which arcs need to be reconsidered?

Arc consistency algorithm (for binary constraints)

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Procedure GAC(V,dom,C)           Inputs                     V: a set of variables                     dom: a function such that dom(X) is the domain of variable X                     C: set of constraints to be satisfied           Output                     arc-consistent domains for each variable           Local                     DX is a set of values for each variable X                    TDA is a set of arcs1:           for each variable X do2:                     DX ←dom(X) 3:           TDA ←{ X,c | X ∈ V, c ∈ C  and X ∈ scope(c)} ⟨ ⟩         4:           while (TDA {}) 5:                     select X,c ∈TDA⟨ ⟩6:                     TDA ←TDA  \ { X,c }⟨ ⟩7:                     NDX ←{x| x ∈ DX and y ∈ DY s.t. (x, y) satisfies c}8:                     if (NDX DX) then 9:                               TDA ←TDA ∪ { Z,c' | X ∈ scope(c'), c' ⟨ ⟩ c, Z ∈ scope(c') \ {X} } 10:                             DX ←NDX                     11:           return {DX| X  is a variable}

Scope of constraint c is the set of variables involved in that constraint

NDX: values x for X for which there a value for y supporting x

X’s domain changed: arcs (Z,c’) for variables Z sharing a constraint c’ with X could become inconsistent

TDA: ToDoArcs,blue arcsin AIspace

Arc Consistency Algorithm: Interpreting Outcomes

• Three possible outcomes (when all arcs are arc consistent):– Each domain has a single value, e.g.

http://www.cs.ubc.ca/~mack/CS322/AIspace/simple-network.xml(Download the file and load it as a local file in AIspace)•We have a (unique) solution.

– At least one domain is empty, e.g.http://www.cs.ubc.ca/~mack/CS322/AIspace/simple-infeasible.xml•No solution! All values are ruled out for this variable.

– Some domains have more than one value, e.g.built-in example “simple problem 2”•There may be a solution, multiple ones, or none•Need to solve this new CSP (usually simpler) problem: same constraints, domains have been reduced

Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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• How often will we prune the domainof variable V? O(d) times

• How many arcs will be put on the ToDoArc list when pruning domain of variable V?- O(degree of variable V)- In total, across all variables: sum of degrees of all variables =

2*number of constraints, i.e. 2*c• Together: we will only put O(dc) arcs on the ToDoArc list• Checking consistency is O(d2) for each of them

• Overall complexity: O(cd3)• Compare to O(dN) of DFS!! Arc consistency is MUCH faster

Arc Consistency Algorithm: Complexity• Worst-case complexity of arc consistency procedure on a

problem with N variables– let d be the max size of a variable domain– let c be the number of constraints

Lecture Overview

• Solving Constraint Satisfaction Problems (CSPs)- Recap: Generate & Test- Recap: Graph search

• Arc consistency- GAC algorithm- Complexity analysis- Domain splitting

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Can we have an arc consistent network with non-empty domains

that has no solution?YES

NO

• Example: vars A, B, C with domain {1, 2} and constraints A ≠ B, B ≠ C, A ≠ C

• Or see AIspace CSP applet Simple Problem 2

Domain splitting (or case analysis)

• Arc consistency ends: Some domains have more than one value may or may not have a solutionA. Apply Depth-First Search with Pruning orB. Split the problem in a number of disjoint cases:

CSP with dom(X) = {x1, x2, x3, x4} becomes

CSP1 with dom(X) = {x1, x2} and CSP2 with dom(X) = {x3, x4}

• Solution to CSP is the union of solutions to CSPi

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Domain splitting• Each smaller CSP is easier to solve

– Arc consistency might already solve it• For each subCSP, which arcs have to be on the

ToDoArcs list when we get the subCSP by splitting the domain of X?arcs <Z, r(Z,X)>

arcs <Z, r(Z,X)> and <X, r(Z,X)>

All arcs

A1 c1

A2 c2

A3 c3

Yc

THESE

X

Ac4

THIS

• Trace it on “simple problem 2”

Domain splitting in action

If domains with multiple values Split on one

Searching by domain splitting

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How many CSPs do we need to keep around at a time? With depth m and 2 children at each split: O(2m). It’s a DFS.

CSP, apply AC

CSP1, apply AC CSP2, apply ACIf domains with multiple values Split on one

If domains with multiple values…..Split on one

Learning Goals for today’s class• Define/read/write/trace/debug the arc

consistency algorithm. Compute its complexity and assess its possible outcomes

• Define/read/write/trace/debug domain splitting and its integration with arc consistency

• Coming up: local search, Section 4.8